Geometry of rays-positive manifolds

TL;DR

This paper investigates rays-positive manifolds, focusing on their geometric properties, classifications, and invariants, including applications to varieties with crepant singularities and bounds on sectional genus.

Contribution

It introduces the concept of rays-positive manifolds, classifies certain varieties with crepant singularities, and proves non-negativity of the sectional genus with characterizations for specific cases.

Findings

Classification of projective varieties with crepant singularities and small degree

Proof of non-negativity of the sectional genus g(M,L)

Characterization of pairs with g(M,L) = 0,1

Abstract

Let M be a smooth complex projective variety and let L be a line bundle on it. Rays-positive manifolds, namely pairs (M,L) such that L is numerically effective and L\cdotR > 0 for all extremal rays R on M, are studied. Several illustrative examples and some applications are provided. In particular, projective varieties with crepant singularities and of small degree with respect to the codimension are classified, and the non-negativity of the sectional genus g(M,L) is proven, describing as well the pairs with g(M,L) = 0,1.